Talk:Itô's lemma

Is anyone who knows what they are doing, re: Wiki, still monitoring this page? I note that Pcb21/Pete seems to have taken a break from Wiki.

In any case, (a) it is incorrect to state Itō's Lemma in "differential form" as is done in this article (and just about everywhere else that the Lemma is "stated") -- the equality holds for the integral, but not for the differential "equivalent;" (b) I can provide a formal proof (my own), if someone is willing to verify it, but I am completely unfamiliar with the math editor here. --Marsden 23:52, 23 August 2005 (UTC)

The differential and integral form are exactly equivalent. What makes you think they differ?

I've recently written up a formal proof to Ito's lemma if anyone wants it.
 * Wikipedia itself would greatly benefit from such a proof. Would you be willing to release under our free licence? If you don't want to spend the time converting from what format you have it in to wiki-markup you could send it to me and I could do it. Let me know! Thanks for your interest Pete/Pcb21 (talk) 08:52, 11 Feb 2004 (UTC)

Shouldn't it be "Itô's lemma" with the appropriate accent mark? I think "Ito's lemma" (without the accent) was the cause of O.J. Simpson's acquittal or something? --Christofurio 15:36, Apr 12, 2004 (UTC)


 * moved, kept redirect. If something is written about the OJ case, we would need to re-think the names to minimize confusion. Pete/Pcb21 (talk) 20:41, 12 Apr 2004 (UTC)


 * What is with the ô? It's a Japanese name and in Romaji, o is o; there is no ô. Is it an Ainu name?

What does the formal proof require? We say it needs different things in two different places... can we rationalize/improve on this? Pete/Pcb21 (talk) 23:32, 15 Apr 2004 (UTC)

I think this "proof" should not really appear on the page, since the Taylor serie does not always exist, that we can't really reorder the terms of the expansion, and finally that the error term may not be negligeable. And finally, this proof does not give the intuition of why the Ito's lemma is true.. 140.247.43.68 20:23, 19 March 2006 (UTC)

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i'm interested in this and other stochastic calc stuff for financial applications, but my background is only differential and integral calculus (called I and II usually), and that was years ago. can someone, please, explain what ito's lemma does or means just in english, without any mathematical symbols? that would be very helpful. thanks guys.

this probably applies to darn near all the math pages, i think. it makes it usable for more people if it has an english-only description (could be even 1 sentence) somewhere in there, preferably at the beginning.

i know you guys are very good with math but some of us plebeians are lost by this sort of notation. :) thanks guys.


 * The differential of a function is a linear approximation of the change in the function. That is, the graph of the function is approximated by a straight line or a plane. If f is a function of t and W, then its differential df is a linear function of dt and dW that is approximately equal to the change in f. The differential df is calculated from the actual change by ignoring higher-order terms, e.g. squares and cubes. However, in stochastic calculus dW^2 is not really a higher-order term, and so it cannot be ignored. The order of dW is really one half so that the order of dW^2 is one; in fact dW^2 is equal to dt. The usual formula for the differential of a function has to be modified to take this into account, and this is what Ito's lemma does.—Zophar 15:59, 20 November 2006 (UTC)


 * If you have a function f(x), the change in f due to a small change in x is df = (df/dx) dx.  For a function of 2 variables g(x,t), a change in g due to small changes in x and t is dg = (dg/dx)dx + (dg/dt)dt.


 * But that's for "sure variables". Variables that have a known value at a known time.  At time 2, the variable t has a value of 2.  We know that for sure.  At time 5, the variable x has a value of x(5), with 100% certainty.


 * There are things called random, or stochastic, variables (meaning a variable we don't know the value of at a given time until that time has occured) we only know the statistics of that variable. For example, w(3) is 10% likely to be between 9 and 10.  Until t=3 happens and w(3) is "realized".  They are also called "processes", but they can also be called "stochastic variables" too.  Brownian motion is an example of such a process.


 * We can have functions of these Brownian motion variables, like f=W^2(t) or f = Log(Sin(gd(arcTanh(1/W(t))))).  For such a function, our differential scheme that I labeled above doesn't work, and there's no reason to expect that it does because we're working with a completely different type of function now.


 * To get the change in this type of f, due to small changes of these stochastic variables, you need to use Ito's Lemma. That's all it is.


 * Your goal is to get the change in f due to small changes in the variables f depends on. For "sure variables", we uses Newton's differential formula (dunno if it has a name).  When f depends on stochastic variables, we use Ito's Lemma.


 * Sliver 19:32, 3 May 2007 (UTC)

Entry Does Not Deliver Promise
The sentence "This is not Ito's Lemma, and is in fact just a specialization of the Lemma." is an odd one. I know this as Ito's Lemma. If it really isn't Ito's Lemma, then either this page should be renamed or Ito's full lemma needs to be stated. We can't have an encyclopedia that purports to have an entry named "Foo" and not describe "Foo". Either this page describes Ito's Lemma or it shouldn't be the Ito's Lemma page.

Can someone please tell me what Ito's Lemma is if this isn't it? Otherwise, I intend to delete this sentence since this is the Ito's lemma I've come to know and love.

If there are no objections, I also intend to change some of the wording so that terms don't "disappear" or get "deleted". Instead, we should state that the calculation is performed to O(delta t).

Sliver 19:32, 3 May 2007 (UTC)

Additional to all previously mentioned, does anyone have the correct form for when Ito's Lemma applies to more than one stochastically defined process? I think it's something in the order of basically doubling up on all of the terms and also including a d^2x/(dydz) value within the dt component. It appears to be coming up frequently in quant finance courses. DC34. —Preceding unsigned comment added by Dc34 (talk • contribs) 11:52, 20 October 2007 (UTC)

I removed the line "This is not Ito's Lemma", as it is a confusing, and very strange thing to say right after the definition of Ito's Lemma. This is clearly a case of Ito's lemma, even if it is not in the most general form. I think the general statement for multidimensional semimartingales should be added. If no-one else does this, then I will do this when I have the time.

Sorry, forgot to sign my previous comment.Roboquant (talk) 14:58, 3 March 2008 (UTC)

Defintion required
Can someone provide an improvement for "(B represents random variation over time)" st the start of the first main section. Melcombe (talk) 14:59, 17 February 2009 (UTC)


 * I have made what I think is an improvement to this --128.243.220.41 (talk) 11:02, 1 May 2009 (UTC)


 * Thanks, that's rather better. Melcombe (talk) 16:22, 1 May 2009 (UTC)

Multi-dimensional formula
Strictly speaking, the multi-dimensional expression for Ito's lemma is not correct. It does hold if you use the identities (not mentioned until much later in the article) dt^2 = 0 and dB^2 = dt, and some other creative manipulations. However, in more than one dimension the Hessian of the function f is a TENSOR not a MATRIX. So it's very misleading to state the formula in this way.

Also, many people have asked for some layperson explanations of Ito's lemma. IMHO, this belongs in the simple english translation of the wiki page, not here.

bradweir (talk) 20:54, 29 May 2012 (UTC)


 * "the simple english translation of the wiki page" is entirely separate. Articles on Wikipedia should try to follow Wikipedia standards, in this case the most relevant is WP:MOSMATH. This would require at least, for the lead, an understandable explanation of what Ito's lemma is about and why/where it is important, but not necessarily an explanation of the lemma itself. It looks like the lead section could do with some work, as "Taylor's expansion" doesn't seem relevant, or not obviously relevant. Melcombe (talk) 11:48, 30 May 2012 (UTC)


 * My main point is that the multi-dimensional formula is incorrect, the comment about lay-speak is tangential. Sorry to attack the messenger, but if you don't understand the connection with Taylor series (which was a primary motivation of Ito's work), then you probably should .  bradweir (talk) 21:49, 21 June 2012 (UTC)


 * I answered your main point since your first point was self-evidently trivial. The function f is clearly scalar and there is nothing in the article to suggest otherwise, so tensors are irrelevant. This part of the article is no worse than any of the rest in terms of things being properly defined, so there is no particular reason to pick it out. Of course we know that mathematicians are unable to write clearly and in this case there are no citations that anyone one can check for what was actually intended .... this is symptomatic of mathematics articles on Wikipedia. We also know that there are many how complain and claim to know better but contribute nothing. Clearly the lead does fail the requirements of WP:LEAD (which is one of the important rquirements of Wikipedia): it goes off on a tangent about "Taylor's expansion" ... even assuming that a lay person would know what might be meant by this, any mathematician would know that that you can't have a "Taylor's expansion" without having something of which make to have a "Taylor's expansion" ... and any indication of this is gloriously absent. Now we have someone who claims to know what "was a primary motivation of Ito's work" but who has failed to include this information in the article, with an approptiate citation. Further they claim to know something about Ito's lemma and that some things in the article are wrong, yet they do nothing about it. So "bradweir", are to we assume that, because you have decided to "avoid making any substantial changes to this article", that you don't understand Ito's lemma ...well if you will adopt a supercilious tone you can expect others to join in. Melcombe (talk) 00:44, 23 June 2012 (UTC)


 * I propose we delete this thread. As I stated earlier, I was not trying to levy a personal attack on you, nor did I wish to start a debate about the exposition of mathematics.


 * I did not find it obvious that the function f is scalar: Ito's lemma applies to vector-valued functions as well. Ito's lemma is a statement about a function, so here we are talking about the Taylor expansion of that function.  I'll change the article to reflect this.  Feel free to remove any of the changes.  bradweir (talk) 21:33, 28 June 2012 (UTC)


 * Your comment has actually raised an important issue ... the treatment of the multivariate case. This article points back to Itō drift-diffusion process (which is itself a redirect) but that doesn't realy cover the question of multivariate processes defined via a stochastic differential equation driven by a multivariate noise. It would seem to require a definition of a multivariate Wiener process, but while Wiener process has some discussion of multivariate Brownian motion, it doesn't seem to define or discuss how to treat a multivariate version of the Wiener process. Condidering stochastic differential equation, this does have some stuff about the multivariate case and does mention "an m-dimensional Brownian motion B" but without giving a pointer to what this might mean. From this, it seems that this article isn't the place to start making the improvements required if the multivariate case is to be properly covered. Melcombe (talk) 11:17, 29 June 2012 (UTC)


 * An N-dimensional Brownian motion/Wiener process is just N iid Brownian motions. This is simple, but you're right in pointing out that many articles just assume the reader knows it bradweir (talk) 22:03, 7 March 2013 (UTC)


 * The Nd formula isnt Ito's lemma at all, it is just a 2nd order taylor expansion. Ito's lemma shows how to incorporate a stochastic drift term of linear order in dt by considering terms quadratic in dW. What is displayed shows a term quadratic in dX. — Preceding unsigned comment added by 31.55.69.189 (talk) 00:14, 23 September 2013 (UTC)


 * Yes, this is a problem, which I was trying to point out in my original post. There are some notational issues, and the fact that I didn't want to fix everything that was wrong. I'll give it a shot. briardew (talk) 20:03, 14 May 2014 (UTC)


 * Ok, this should be improved now. It still needs a lot of work, but we've come a long way from where it started. briardew (talk) 17:12, 1 July 2014 (UTC)

First Example
Which version of Ito’s Lemma is Geometric Brownian Motion meant to be an example of? It looks like an application of the case of an Ito drift-diffusion process with sigma_t = S*sigma. But doesn’t the notation sigma_t imply that sigma_t is a function of only t, not S? The page for Geometric Brownian Motion seems to essentially re-derive the lemma for its special case. Tnedde (talk) 18:50, 19 February 2019 (UTC)Tnedde


 * Yes, but S is a function of t... Hairer (talk) 11:21, 20 February 2019 (UTC)

Requested move 13 June 2019

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 

The result of the move request was: After relisting, no obvious strong preference or reason to prefer one of the common names over the other; so no move. (non-admin closure) Dicklyon (talk) 05:46, 4 July 2019 (UTC)

Itô's lemma → Itô's formula – The existing literature usually refers to it as "Itô's formula" rather than "Itô's lemma" (see the standard textbooks by Revuz & Yor, Mörters & Peres, etc. Hairer (talk) 08:10, 13 June 2019 (UTC) --Relisting. DannyS712 (talk) 20:07, 20 June 2019 (UTC)
 * Oppose - I have always seen this in textbooks as "Itô's lemma", for example, . Rlendog (talk) 14:24, 13 June 2019 (UTC)
 * The classic textbooks written by the generation of mathematicians who knew Itô personally all refer to it as "Itô's formula", see Ikeda-Watanabe Sec 2.5, Revuz-Yor Sec IV.3, Stroock Sec 7.2.3., Rogers-Williams Sec IV.3. Hairer (talk) 11:07, 15 June 2019 (UTC)


 * Support Jarrow Protter (‘A short history of stochastic integration..’ etc): “The book by H. P. McKean, Jr., published in 1969, had a great influence in popularizing the Itˆo integral, as it was the first explanation of Itˆo’s and others’ related work in book form. But McKean referred to Itˆo’s formula as Itˆo’s lemma, a nomenclature that has persisted in some circles to this day. Obviously this key theorem of Itˆo is much more important than the status the lowly nomenclature “lemma” affords it, and we prefer Itˆo’s own description: “formula”.” My emphasis. IFRS17 (talk) 10:27, 17 June 2019 (UTC)
 * I just found Equity Derivatives by Overhaus et al on my bookshelf. p. 9 states "Ito's formula - We can now state the famous Ito's formula" IFRS17 (talk) 11:49, 17 June 2019 (UTC)
 * See also Brigo & Mercurio Interest rate models Theory and Practice p.477. IFRS17 (talk) 12:08, 17 June 2019 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Structure of page
Is there a reason that an informal derivation of the lemma appears before the statement of the lemma itself? I found this confusing. Would anyone object if I moved the statement to before the informal derivation? 129.215.104.157 (talk) 16:50, 26 September 2019 (UTC)


 * I might personally find an informal derivation before the formal one to be more helpful, as part of the motivation/intuition, but this mostly seems like a matter of preference. That both are here is good, and I don't imagine there is a 'right' order. Youarelovedfool (talk) 03:31, 20 May 2023 (UTC)